A key question in condensed-matter physics is to understand how high-temperature superconductivity emerges on adding mobile charged carriers to an antiferromagnetic Mott insulator. We address this question using angle-resolved photoemission spectroscopy to probe the electronic excitations of the non-superconducting state that exists between the Mott insulator and the d-wave superconductor in Bi{sub 2}Sr{sub 2}CaCu{sub 2}O{sub 8+{delta}}. Despite a temperature-dependent resistivity characteristic of an insulator, the excitations in this intermediate state have a highly anisotropic energy gap that vanishes at four points in momentum space. This nodal-liquid state has the same gap structure as that of the d-wave superconductor but no sharp quasiparticle peaks. We observe a smooth evolution of the excitation spectrum, along with the appearance of coherent quasiparticles, as one goes through the insulator-to-superconductor transition as a function of doping. Our results suggest that high-temperature superconductivity emerges when quantum phase coherence is established in a non-superconducting nodal liquid.